Question

Please show me how to solve the initial value problem
y'=tanx y(pi/4)=3

Answer 1

The ODE is separable, i.e. you can write


`(\mathrm dy)/(\mathrm dx)=\tan x\iff\mathrm dy=\tan x\,\mathrm dx`

Integrating both sides gives the general solution.


`\displaystyle\int\mathrm dy=\int\tan x\,\mathrm dx`

`y=-\ln|\cos x|+C`

Given that
`y\left(\frac\pi4\right)=3`, we have


`3=-\ln\left|\cos\frac\pi4\right|+C`

`3=-\ln\frac1{\sqrt2}+C`

`3-\ln\sqrt2=C`


and so the particular solution to the IVP is


`y=-\ln|\cos x|+3-\ln\sqrt2`

`y=3-\ln|\sqrt2\cos x|`